The Reincarnated Science Whiz

Chapter 24: The First Day of Competition

In 2009, coinciding with the 50th anniversary of the International Mathematical Olympiad (IMO), the International Mathematical Olympiad Committee held a 50th-anniversary celebration. Many world-renowned mathematicians attended this 50th-anniversary celebration. Following the celebration, the official competition began, with nearly 560 students from 105 countries and regions around the world participating. The entire competition lasted one week. During this week, the contestants would tackle challenging math problems, competing for the gold, silver, and bronze medals of the International Mathematical Olympiad. Contestants from every country arrived determined to bring honor to their nations on the world stage. On March 15, the competition kicked off The IMO consists of six problems in total—three will be administered today and three tomorrow—with each problem worth 7 points, for a maximum score of 42 points. The competition lasts 4. 5 hours each day. Contestants may bring any writing utensils and drawing tools, but no electronic devices are permitted in the exam room. Because the competition lasts a long time, contestants may bring their own food and drinks into the venue and may carry no more than three reference materials. However, apart from some food and drinks, Qin Yuanqing didn’t bring a single reference book, because based on past experience, reference materials are essentially useless—the problem-setters have already accounted for them. If a solution could be found in a reference book, it would mean the problem-setters’ skills were subpar. This is similar to exams in China, where open-book exams are often much more difficult than closed-book exams. Since the questions provided to local contestants had already been translated into their native language, the contestants faced no language barriers when receiving their exam papers. When Qin Yuanqing received the exam paper, there were only three questions. The first one was the simplest; if he couldn’t even solve the first question, there was no point in even considering the next two. Qin Yuanqing remained calm. The first problem was the simplest—a gimme—but at the same time, one wrong move could turn it into a deal-breaker. “1. Let n be a positive integer, and let a₁, a₂, …, a_k (k ≥ 2) be distinct integers from {1, 2, …, n}, such that n | a_i(a_i + 1 − 1) holds for all i = 1, 2, …, k − 1. Prove that a_k(a₁ − 1) is not divisible by n.” Qin Yuanqing read the problem three times, silently cursing the person who provided it—wishing they’d be born without an anus—for setting such a trap; one wrong move and you’d get the answer wrong. Qin Yuanqing began his solution. First, he used mathematical induction to prove that for any integer i (2 ≤ i ≤ k), the expression is divisible by n. He concluded that when i = 2, the given condition—that the expression is divisible by n—holds true. Proceeding step by step in this manner, he finally reached the conclusion that ak(a1 - 1) is not divisible by n. Qin Yuanqing then turned his attention to the second problem. “Let O be the center of the circumcircle of △ABC. Points P and Q lie on line segments CA and AB, respectively. Points K, L, and M are the midpoints of BP, CQ, and PQ, respectively. Circle Г passes through points K, L, and M and is tangent to PQ. Prove that OP = OQ.” After carefully reading this problem, Qin Yuanqing felt it was somewhat easier than the previous one and did not contain any traps. He first drew a circle, then transformed it into △ABC, and next constructed the line segments CA and AB as well as points P and Q. He then marked the midpoints K, L, and M of BP, CQ, and PQ, respectively. Finally, he constructed circle Г. Next, he drew line PQ so that it was tangent to circle Γ at point M. Using the chord-tangent angle theorem, he deduced that ∠QMK = ∠MLK. Since points K and M are the midpoints of BP and PQ, respectively, KM is parallel to BQ, which implies that ∠QMK = ∠AQP. Therefore, ∠MLK = ∠AQP. By the same reasoning, ∠MKL = ∠APQ. By the equality of angles, we have △MKL ∽ △APO, so MK/ML = AP/AQ Since K, L, and M are the midpoints of line segments BP, CQ, and PQ, respectively, we have KM = BQ/2 and LM = CP/2. Substituting these into the above equation gives BQ/CP = AP/AQ. Rearranging the equation yields AP·CP = AQ·BQ. By the Pythagorean theorem, we have OP² = OA² − AP·CP = OA² − AQ·BQ = OQ². Therefore, we conclude that OP = OQ. Qin Yuanqing didn’t even check his work; he transformed the abstract math problem into a graphical representation—this is where he excels, and he was absolutely certain he could prove it. Qin Yuanqing immediately turned his attention to the third problem: “3. S₁, S₂, S₃, … is a strictly increasing sequence of positive integers, and its subsequences S_(S1), S_(S2), S_(S3), … and S_(S1+1), S_(S2+1), S_(S3+1), … are all arithmetic sequences. Prove that S₁, S₂, S₃, … is an arithmetic sequence.” Looking at this problem, Qin Yuanqing furrowed his brow slightly; it was clearly much harder than the previous two. He briefly reviewed the given conditions and realized that this problem combined arithmetic sequences with the transformation method. Qin Yuanqing worked through the problem step by step. Since both the sequence and its subsequences consist of strictly increasing positive integers, he let Ssk = a + (k-1)d1 and SSk+1 = b + (k-1)d2 (k = 1, 2, ... , where a, b, d1, and d2 ∈ N+). After reformulating the problem in terms of functions and sequences, and given that $S_k < S_{k+1} < S_{k+1}$ and the monotonicity of $\{S_n\}$, we know that for any positive integer $k$, $S_k < S_{k+1} \leq S_{k+1}$. That is, $a + (k-1)d_1 < b + (k-1)d_2 \leq a + kd_1 Therefore, a − b ≤ (k − 1)(d₂ − d₁) ≤ a + d₁ − b. Since k is arbitrary, we know that d₂ − d₁ = 0, so d₂ = d₁. . . . When Qin Yuanqing wrote down the conclusion of his proof, he touched his forehead and found that he was already sweating; he let out a soft sigh. Qin Yuanqing then stood up and signaled that he was finished. The proctor walked over to him, placed his exam paper in a sealed envelope, and sealed it. Qin Yuanqing left the exam room with ease and composure, feeling no pressure at all. Since he had completed his answers, there could be no mistakes. It wasn’t until Qin Yuanqing left the exam room that he realized he was the first to turn in his paper; none of the members of the Huaxia Math Olympiad Team had turned theirs in yet, nor had a single member of any other country’s Math Olympiad team. “How did the first day of the competition go?” the assistant team leader asked as soon as he saw Qin Yuanqing. “Just so-so—it was a breeze!” Qin Yuanqing waved his hand nonchalantly. “It wasn’t as hard as the training camp exams. Don’t worry—I’m definitely going to score at least 42 points!” Hearing this, the assistant team leader immediately breathed a sigh of relief. On the Huaxia Math Olympiad Team, Qin Yuanqing was the ace and the rock of the team; if he said so, it meant this year’s exam wasn’t too difficult. “It was just the first problem—I don’t know which country set it—but they set a trap. If you’re not careful, you’ll get it wrong. That’s just too underhanded—playing mind games with us high school students!” Qin Yuanqing complained. Just then, Qin Yuanqing noticed a tall, burly white man not far away staring at them with a hostile glare. The assistant team leader quickly covered Qin Yuanqing’s mouth with his hand and whispered, “I heard the first problem was set by Australia. That guy is the assistant team leader of the Australian Math Olympiad team!” Qin Yuanqing was speechless. He’d been badmouthing someone behind their back, and the person had actually heard him—that really reflected poorly on his character. But as soon as he heard it was Australia, he instantly thought the country was downright unscrupulous. Even before his rebirth, Australia had lost its mind for some reason, constantly butting heads with Huaxia, which had led to a flood of online criticism. Now, the other two problems were perfectly normal—especially the last one, which was very well crafted—but they’d resorted to underhanded tricks on the very first problem. Australia really wasn’t in its right mind. Qin Yuanqing just couldn’t figure it out: with Australia being so brain-dead, how come so many Huaxia people still immigrate there, only to have their own minds corrupted in the process? Take that infamous Liang Mouyan from early 2020, for example—she was arrogant and unreasonable, even screaming for help and claiming someone was harassing her. If it weren’t for the video evidence, it would’ve been impossible to prove she was in the wrong. After being deported, she actually demanded that people from Huaxia apologize to her and reimburse her flight ticket—she’s clearly lost her mind. About half an hour later, some Indians walked out of the exam hall. Qin Yuanqing asked curiously, “Deputy Team Leader, are Indians really that good at math?” The deputy team leader replied, “Of course. India ranks among the best in the world in mathematics, second only to the Ramanujan Prize—which is named after the Indian mathematician Ramanu.” “Ramanujan was quite a genius; the Ramanujan Conjectures are among the most formidable,” Qin Yuanqing nodded slightly. As for the contestants from Russia who emerged next, mathematics had been exceptionally strong during the period of the Soviet Union, producing many great mathematicians. with figures like Sergei and Derifeld having won the Fields Medal. Russia, having inherited most of the legacy of the Soviet Union, is naturally very strong in mathematics as well. For example, Grigori Perelman is a true genius who solved the Poincaré Conjecture; because of his proof, thousands of related mathematical conjectures were established as theorems, effectively advancing the historical progress of geometry and topology single-handedly. Even though Perelman is an eccentric who dislikes giving interviews and avoids the public eye, there is no doubt that he is undoubtedly one of the greatest mathematicians in the world today. Once the entire Huaxia Math Olympiad Team had gathered, they returned to the hotel together. No one would check the answers—that would only interfere with the next day’s competition. Back at their base, Qin Yuanqing went online and searched for the world’s leading mathematical powers. The United States, Europe, Russia, and Japan all ranked among them, each having produced more than one Fields Medalist. The United States, in particular, stood as the undisputed leader in mathematics—whether judged by the rankings of university mathematics programs, the number of research institutes, or the number of specialized mathematics journals. As for Huaxia, although it had frequently won IMO gold medals over the past decade, it could not be considered a mathematical powerhouse; at best, it was a major player in mathematics. Qin Yuanqing also reflected on what he had seen and heard while out and about a few days earlier: ordinary people in Europe and the United States have extremely poor computational skills, yet their education systems focus on fostering children’s interests. Mathematics is a discipline that demands talent and logical thinking; the bar is set very high. Without mathematical aptitude or sufficient logical reasoning, one simply cannot even enter the field of mathematics. On the other hand, those who are genuinely interested—because it is something they are passionate about—often possess strong self-learning abilities, and when pursuing a subject they love, they tend to achieve twice the results with half the effort. Similarly, the cultivation of mathematical thinking is also crucial. Huaxia is a country with a massive population and a system of nine years of compulsory education that prioritizes fairness and equity. This necessitates a vast teaching force, and for Huaxia, meeting quantity comes first, followed by quality. Consequently, the educational process relies on a “cram-school” style of teaching. Students educated this way tend to achieve similar exam scores, but their critical thinking skills remain a significant issue. By the time they reach college, the gap in mathematical ability between Huaxia University students and their counterparts abroad becomes apparent. Mathematical prodigies abroad receive excellent training in mathematical thinking from a young age; with a solid foundation built over time, they demonstrate exceptional ability by the time they reach college. Overseas education is elite-oriented, while domestic education is geared toward the masses; the disparity in educational systems inevitably leads to different outcomes. Meanwhile, the domestic education system has produced millions of engineers—a high-quality yet inexpensive engineering workforce—to the point that by around 2018, the “engineer dividend” had become a new buzzword.